--- a/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,155 +7,162 @@
theory HahnBanachExtLemmas = FunctionNorm:
-text{* In this section the following context is presumed.
-Let $E$ be a real vector space with a
-seminorm $q$ on $E$. $F$ is a subspace of $E$ and $f$ a linear
-function on $F$. We consider a subspace $H$ of $E$ that is a
-superspace of $F$ and a linear form $h$ on $H$. $H$ is a not equal
-to $E$ and $x_0$ is an element in $E \backslash H$.
-$H$ is extended to the direct sum $H' = H + \idt{lin}\ap x_0$, so for
-any $x\in H'$ the decomposition of $x = y + a \mult x$
-with $y\in H$ is unique. $h'$ is defined on $H'$ by
-$h'\ap x = h\ap y + a \cdot \xi$ for a certain $\xi$.
+text {*
+ In this section the following context is presumed. Let @{text E} be
+ a real vector space with a seminorm @{text q} on @{text E}. @{text
+ F} is a subspace of @{text E} and @{text f} a linear function on
+ @{text F}. We consider a subspace @{text H} of @{text E} that is a
+ superspace of @{text F} and a linear form @{text h} on @{text
+ H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is
+ an element in @{text "E - H"}. @{text H} is extended to the direct
+ sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"}
+ the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is
+ unique. @{text h'} is defined on @{text H'} by
+ @{text "h' x = h y + a \<cdot> \<xi>"} for a certain @{text \<xi>}.
-Subsequently we show some properties of this extension $h'$ of $h$.
-*}
-
+ Subsequently we show some properties of this extension @{text h'} of
+ @{text h}.
+*}
-text {* This lemma will be used to show the existence of a linear
-extension of $f$ (see page \pageref{ex-xi-use}).
-It is a consequence
-of the completeness of $\bbbR$. To show
-\begin{matharray}{l}
-\Ex{\xi}{\All {y\in F}{a\ap y \leq \xi \land \xi \leq b\ap y}}
-\end{matharray}
-it suffices to show that
-\begin{matharray}{l} \All
-{u\in F}{\All {v\in F}{a\ap u \leq b \ap v}}
-\end{matharray} *}
+text {*
+ This lemma will be used to show the existence of a linear extension
+ of @{text f} (see page \pageref{ex-xi-use}). It is a consequence of
+ the completeness of @{text \<real>}. To show
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}
+ \end{tabular}
+ \end{center}
+ \noindent it suffices to show that
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"}
+ \end{tabular}
+ \end{center}
+*}
-lemma ex_xi:
- "[| is_vectorspace F; !! u v. [| u \<in> F; v \<in> F |] ==> a u <= b v |]
- ==> \<exists>xi::real. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"
+lemma ex_xi:
+ "is_vectorspace F \<Longrightarrow> (\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v)
+ \<Longrightarrow> \<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
proof -
assume vs: "is_vectorspace F"
- assume r: "(!! u v. [| u \<in> F; v \<in> F |] ==> a u <= (b v::real))"
+ assume r: "(\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> (b v::real))"
txt {* From the completeness of the reals follows:
- The set $S = \{a\: u\dt\: u\in F\}$ has a supremum, if
+ The set @{text "S = {a u. u \<in> F}"} has a supremum, if
it is non-empty and has an upper bound. *}
let ?S = "{a u :: real | u. u \<in> F}"
- have "\<exists>xi. isLub UNIV ?S xi"
+ have "\<exists>xi. isLub UNIV ?S xi"
proof (rule reals_complete)
-
- txt {* The set $S$ is non-empty, since $a\ap\zero \in S$: *}
- from vs have "a 0 \<in> ?S" by force
+ txt {* The set @{text S} is non-empty, since @{text "a 0 \<in> S"}: *}
+
+ from vs have "a 0 \<in> ?S" by blast
thus "\<exists>X. X \<in> ?S" ..
- txt {* $b\ap \zero$ is an upper bound of $S$: *}
+ txt {* @{text "b 0"} is an upper bound of @{text S}: *}
- show "\<exists>Y. isUb UNIV ?S Y"
- proof
+ show "\<exists>Y. isUb UNIV ?S Y"
+ proof
show "isUb UNIV ?S (b 0)"
proof (intro isUbI setleI ballI)
show "b 0 \<in> UNIV" ..
next
- txt {* Every element $y\in S$ is less than $b\ap \zero$: *}
+ txt {* Every element @{text "y \<in> S"} is less than @{text "b 0"}: *}
- fix y assume y: "y \<in> ?S"
+ fix y assume y: "y \<in> ?S"
from y have "\<exists>u \<in> F. y = a u" by fast
- thus "y <= b 0"
+ thus "y \<le> b 0"
proof
- fix u assume "u \<in> F"
+ fix u assume "u \<in> F"
assume "y = a u"
- also have "a u <= b 0" by (rule r) (simp!)+
+ also have "a u \<le> b 0" by (rule r) (simp!)+
finally show ?thesis .
qed
qed
qed
qed
- thus "\<exists>xi. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"
+ thus "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
proof (elim exE)
- fix xi assume "isLub UNIV ?S xi"
+ fix xi assume "isLub UNIV ?S xi"
show ?thesis
- proof (intro exI conjI ballI)
-
- txt {* For all $y\in F$ holds $a\ap y \leq \xi$: *}
-
+ proof (intro exI conjI ballI)
+
+ txt {* For all @{text "y \<in> F"} holds @{text "a y \<le> \<xi>"}: *}
+
fix y assume y: "y \<in> F"
- show "a y <= xi"
- proof (rule isUbD)
+ show "a y \<le> xi"
+ proof (rule isUbD)
show "isUb UNIV ?S xi" ..
- qed (force!)
+ qed (blast!)
next
- txt {* For all $y\in F$ holds $\xi\leq b\ap y$: *}
+ txt {* For all @{text "y \<in> F"} holds @{text "\<xi> \<le> b y"}: *}
fix y assume "y \<in> F"
- show "xi <= b y"
+ show "xi \<le> b y"
proof (intro isLub_le_isUb isUbI setleI)
show "b y \<in> UNIV" ..
- show "\<forall>ya \<in> ?S. ya <= b y"
+ show "\<forall>ya \<in> ?S. ya \<le> b y"
proof
fix au assume au: "au \<in> ?S "
hence "\<exists>u \<in> F. au = a u" by fast
- thus "au <= b y"
+ thus "au \<le> b y"
proof
- fix u assume "u \<in> F" assume "au = a u"
- also have "... <= b y" by (rule r)
+ fix u assume "u \<in> F" assume "au = a u"
+ also have "... \<le> b y" by (rule r)
finally show ?thesis .
qed
qed
- qed
+ qed
qed
qed
qed
-text{* \medskip The function $h'$ is defined as a
-$h'\ap x = h\ap y + a\cdot \xi$ where $x = y + a\mult \xi$
-is a linear extension of $h$ to $H'$. *}
+text {*
+ \medskip The function @{text h'} is defined as a
+ @{text "h' x = h y + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a
+ linear extension of @{text h} to @{text H'}. *}
-lemma h'_lf:
- "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in h y + a * xi);
- H' == H + lin x0; is_subspace H E; is_linearform H h; x0 \<notin> H;
- x0 \<in> E; x0 \<noteq> 0; is_vectorspace E |]
- ==> is_linearform H' h'"
+lemma h'_lf:
+ "h' \<equiv> \<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi
+ \<Longrightarrow> H' \<equiv> H + lin x0 \<Longrightarrow> is_subspace H E \<Longrightarrow> is_linearform H h \<Longrightarrow> x0 \<notin> H
+ \<Longrightarrow> x0 \<in> E \<Longrightarrow> x0 \<noteq> 0 \<Longrightarrow> is_vectorspace E
+ \<Longrightarrow> is_linearform H' h'"
proof -
- assume h'_def:
- "h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
+ assume h'_def:
+ "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
in h y + a * xi)"
- and H'_def: "H' == H + lin x0"
- and vs: "is_subspace H E" "is_linearform H h" "x0 \<notin> H"
- "x0 \<noteq> 0" "x0 \<in> E" "is_vectorspace E"
+ and H'_def: "H' \<equiv> H + lin x0"
+ and vs: "is_subspace H E" "is_linearform H h" "x0 \<notin> H"
+ "x0 \<noteq> 0" "x0 \<in> E" "is_vectorspace E"
- have h': "is_vectorspace H'"
+ have h': "is_vectorspace H'"
proof (unfold H'_def, rule vs_sum_vs)
show "is_subspace (lin x0) E" ..
- qed
+ qed
show ?thesis
proof
- fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
+ fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
- txt{* We now have to show that $h'$ is additive, i.~e.\
- $h' \ap (x_1\plus x_2) = h'\ap x_1 + h'\ap x_2$
- for $x_1, x_2\in H$. *}
+ txt {* We now have to show that @{text h'} is additive, i.~e.\
+ @{text "h' (x\<^sub>1 + x\<^sub>2) = h' x\<^sub>1 + h' x\<^sub>2"} for
+ @{text "x\<^sub>1, x\<^sub>2 \<in> H"}. *}
- have x1x2: "x1 + x2 \<in> H'"
- by (rule vs_add_closed, rule h')
- from x1
- have ex_x1: "\<exists>y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H"
+ have x1x2: "x1 + x2 \<in> H'"
+ by (rule vs_add_closed, rule h')
+ from x1
+ have ex_x1: "\<exists>y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
- from x2
- have ex_x2: "\<exists>y2 a2. x2 = y2 + a2 \<cdot> x0 \<and> y2 \<in> H"
+ from x2
+ have ex_x2: "\<exists>y2 a2. x2 = y2 + a2 \<cdot> x0 \<and> y2 \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
- from x1x2
+ from x1x2
have ex_x1x2: "\<exists>y a. x1 + x2 = y + a \<cdot> x0 \<and> y \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
@@ -164,181 +171,178 @@
proof (elim exE conjE)
fix y1 y2 y a1 a2 a
assume y1: "x1 = y1 + a1 \<cdot> x0" and y1': "y1 \<in> H"
- and y2: "x2 = y2 + a2 \<cdot> x0" and y2': "y2 \<in> H"
- and y: "x1 + x2 = y + a \<cdot> x0" and y': "y \<in> H"
+ and y2: "x2 = y2 + a2 \<cdot> x0" and y2': "y2 \<in> H"
+ and y: "x1 + x2 = y + a \<cdot> x0" and y': "y \<in> H"
txt {* \label{decomp-H-use}*}
- have ya: "y1 + y2 = y \<and> a1 + a2 = a"
+ have ya: "y1 + y2 = y \<and> a1 + a2 = a"
proof (rule decomp_H')
- show "y1 + y2 + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0"
+ show "y1 + y2 + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0"
by (simp! add: vs_add_mult_distrib2 [of E])
show "y1 + y2 \<in> H" ..
qed
have "h' (x1 + x2) = h y + a * xi"
- by (rule h'_definite)
- also have "... = h (y1 + y2) + (a1 + a2) * xi"
+ by (rule h'_definite)
+ also have "... = h (y1 + y2) + (a1 + a2) * xi"
by (simp add: ya)
- also from vs y1' y2'
- have "... = h y1 + h y2 + a1 * xi + a2 * xi"
- by (simp add: linearform_add [of H]
+ also from vs y1' y2'
+ have "... = h y1 + h y2 + a1 * xi + a2 * xi"
+ by (simp add: linearform_add [of H]
real_add_mult_distrib)
- also have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
+ also have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
by simp
also have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
- also have "h y2 + a2 * xi = h' x2"
+ also have "h y2 + a2 * xi = h' x2"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
-
- txt{* We further have to show that $h'$ is multiplicative,
- i.~e.\ $h'\ap (c \mult x_1) = c \cdot h'\ap x_1$
- for $x\in H$ and $c\in \bbbR$.
- *}
- next
- fix c x1 assume x1: "x1 \<in> H'"
+ txt {* We further have to show that @{text h'} is multiplicative,
+ i.~e.\ @{text "h' (c \<cdot> x\<^sub>1) = c \<cdot> h' x\<^sub>1"} for @{text "x \<in> H"}
+ and @{text "c \<in> \<real>"}. *}
+
+ next
+ fix c x1 assume x1: "x1 \<in> H'"
have ax1: "c \<cdot> x1 \<in> H'"
by (rule vs_mult_closed, rule h')
- from x1
- have ex_x: "!! x. x\<in> H' ==> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
+ from x1
+ have ex_x: "\<And>x. x\<in> H' \<Longrightarrow> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
from x1 have ex_x1: "\<exists>y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
with ex_x [of "c \<cdot> x1", OF ax1]
- show "h' (c \<cdot> x1) = c * (h' x1)"
+ show "h' (c \<cdot> x1) = c * (h' x1)"
proof (elim exE conjE)
- fix y1 y a1 a
+ fix y1 y a1 a
assume y1: "x1 = y1 + a1 \<cdot> x0" and y1': "y1 \<in> H"
- and y: "c \<cdot> x1 = y + a \<cdot> x0" and y': "y \<in> H"
+ and y: "c \<cdot> x1 = y + a \<cdot> x0" and y': "y \<in> H"
- have ya: "c \<cdot> y1 = y \<and> c * a1 = a"
- proof (rule decomp_H')
- show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0"
+ have ya: "c \<cdot> y1 = y \<and> c * a1 = a"
+ proof (rule decomp_H')
+ show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0"
by (simp! add: vs_add_mult_distrib1)
show "c \<cdot> y1 \<in> H" ..
qed
- have "h' (c \<cdot> x1) = h y + a * xi"
- by (rule h'_definite)
+ have "h' (c \<cdot> x1) = h y + a * xi"
+ by (rule h'_definite)
also have "... = h (c \<cdot> y1) + (c * a1) * xi"
by (simp add: ya)
- also from vs y1' have "... = c * h y1 + c * a1 * xi"
- by (simp add: linearform_mult [of H])
- also from vs y1' have "... = c * (h y1 + a1 * xi)"
- by (simp add: real_add_mult_distrib2 real_mult_assoc)
- also have "h y1 + a1 * xi = h' x1"
+ also from vs y1' have "... = c * h y1 + c * a1 * xi"
+ by (simp add: linearform_mult [of H])
+ also from vs y1' have "... = c * (h y1 + a1 * xi)"
+ by (simp add: real_add_mult_distrib2 real_mult_assoc)
+ also have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
qed
qed
-text{* \medskip The linear extension $h'$ of $h$
-is bounded by the seminorm $p$. *}
+text {* \medskip The linear extension @{text h'} of @{text h}
+is bounded by the seminorm @{text p}. *}
lemma h'_norm_pres:
- "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in h y + a * xi);
- H' == H + lin x0; x0 \<notin> H; x0 \<in> E; x0 \<noteq> 0; is_vectorspace E;
- is_subspace H E; is_seminorm E p; is_linearform H h;
- \<forall>y \<in> H. h y <= p y;
- \<forall>y \<in> H. - p (y + x0) - h y <= xi \<and> xi <= p (y + x0) - h y |]
- ==> \<forall>x \<in> H'. h' x <= p x"
-proof
- assume h'_def:
- "h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
+ "h' \<equiv> \<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi
+ \<Longrightarrow> H' \<equiv> H + lin x0 \<Longrightarrow> x0 \<notin> H \<Longrightarrow> x0 \<in> E \<Longrightarrow> x0 \<noteq> 0 \<Longrightarrow> is_vectorspace E
+ \<Longrightarrow> is_subspace H E \<Longrightarrow> is_seminorm E p \<Longrightarrow> is_linearform H h
+ \<Longrightarrow> \<forall>y \<in> H. h y \<le> p y
+ \<Longrightarrow> \<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y
+ \<Longrightarrow> \<forall>x \<in> H'. h' x \<le> p x"
+proof
+ assume h'_def:
+ "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
in (h y) + a * xi)"
- and H'_def: "H' == H + lin x0"
- and vs: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" "is_vectorspace E"
- "is_subspace H E" "is_seminorm E p" "is_linearform H h"
- and a: "\<forall>y \<in> H. h y <= p y"
- presume a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya <= xi"
- presume a2: "\<forall>ya \<in> H. xi <= p (ya + x0) - h ya"
- fix x assume "x \<in> H'"
- have ex_x:
- "!! x. x \<in> H' ==> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
+ and H'_def: "H' \<equiv> H + lin x0"
+ and vs: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" "is_vectorspace E"
+ "is_subspace H E" "is_seminorm E p" "is_linearform H h"
+ and a: "\<forall>y \<in> H. h y \<le> p y"
+ presume a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
+ presume a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya"
+ fix x assume "x \<in> H'"
+ have ex_x:
+ "\<And>x. x \<in> H' \<Longrightarrow> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
by (unfold H'_def vs_sum_def lin_def) fast
have "\<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
by (rule ex_x)
- thus "h' x <= p x"
+ thus "h' x \<le> p x"
proof (elim exE conjE)
fix y a assume x: "x = y + a \<cdot> x0" and y: "y \<in> H"
have "h' x = h y + a * xi"
by (rule h'_definite)
- txt{* Now we show
- $h\ap y + a \cdot \xi\leq p\ap (y\plus a \mult x_0)$
- by case analysis on $a$. *}
+ txt {* Now we show @{text "h y + a \<cdot> \<xi> \<le> p (y + a \<cdot> x\<^sub>0)"}
+ by case analysis on @{text a}. *}
- also have "... <= p (y + a \<cdot> x0)"
+ also have "... \<le> p (y + a \<cdot> x0)"
proof (rule linorder_cases)
- assume z: "a = #0"
+ assume z: "a = #0"
with vs y a show ?thesis by simp
- txt {* In the case $a < 0$, we use $a_1$ with $\idt{ya}$
- taken as $y/a$: *}
+ txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}
+ with @{text ya} taken as @{text "y / a"}: *}
next
assume lz: "a < #0" hence nz: "a \<noteq> #0" by simp
- from a1
- have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) <= xi"
+ from a1
+ have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi"
by (rule bspec) (simp!)
- txt {* The thesis for this case now follows by a short
- calculation. *}
- hence "a * xi <= a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
+ txt {* The thesis for this case now follows by a short
+ calculation. *}
+ hence "a * xi \<le> a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
by (rule real_mult_less_le_anti [OF lz])
- also
+ also
have "... = - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
by (rule real_mult_diff_distrib)
- also from lz vs y
+ also from lz vs y
have "- a * (p (inverse a \<cdot> y + x0)) = p (a \<cdot> (inverse a \<cdot> y + x0))"
by (simp add: seminorm_abs_homogenous abs_minus_eqI2)
also from nz vs y have "... = p (y + a \<cdot> x0)"
by (simp add: vs_add_mult_distrib1)
also from nz vs y have "a * (h (inverse a \<cdot> y)) = h y"
by (simp add: linearform_mult [symmetric])
- finally have "a * xi <= p (y + a \<cdot> x0) - h y" .
+ finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
- hence "h y + a * xi <= h y + p (y + a \<cdot> x0) - h y"
+ hence "h y + a * xi \<le> h y + p (y + a \<cdot> x0) - h y"
by (simp add: real_add_left_cancel_le)
thus ?thesis by simp
- txt {* In the case $a > 0$, we use $a_2$ with $\idt{ya}$
- taken as $y/a$: *}
+ txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}
+ with @{text ya} taken as @{text "y / a"}: *}
- next
+ next
assume gz: "#0 < a" hence nz: "a \<noteq> #0" by simp
- from a2 have "xi <= p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)"
+ from a2 have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)"
by (rule bspec) (simp!)
txt {* The thesis for this case follows by a short
calculation: *}
- with gz
- have "a * xi <= a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
+ with gz
+ have "a * xi \<le> a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
by (rule real_mult_less_le_mono)
also have "... = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"
- by (rule real_mult_diff_distrib2)
- also from gz vs y
+ by (rule real_mult_diff_distrib2)
+ also from gz vs y
have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"
by (simp add: seminorm_abs_homogenous abs_eqI2)
also from nz vs y have "... = p (y + a \<cdot> x0)"
by (simp add: vs_add_mult_distrib1)
also from nz vs y have "a * h (inverse a \<cdot> y) = h y"
- by (simp add: linearform_mult [symmetric])
- finally have "a * xi <= p (y + a \<cdot> x0) - h y" .
-
- hence "h y + a * xi <= h y + (p (y + a \<cdot> x0) - h y)"
+ by (simp add: linearform_mult [symmetric])
+ finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
+
+ hence "h y + a * xi \<le> h y + (p (y + a \<cdot> x0) - h y)"
by (simp add: real_add_left_cancel_le)
thus ?thesis by simp
qed
also from x have "... = p x" by simp
finally show ?thesis .
qed
-qed blast+
+qed blast+
end